Internationally, there is increasing emphasis on teacher leadership of professional development. This provides opportunities for teachers to initiate and facilitate professional learning activities beyond their own schools. There is a need for theoretical tools to analyse their leadership activity and how to support it. Constructs from complexity leadership theory and the concept of teacher system leadership are used to develop a framework to analyse the purposes and practices of teacher professional development leaders supported by a national programme for mathematics teacher professional development in England. I argue that the teachers' activities constitute a form of adaptive leadership involving innovating and organising professional development within arenas of leadership, through the processes of mobilising, brokering and the creation of networks. This required engaging in "system work" to fulfil purposes connected to both local and system-wide concerns. The teachers were supported by the enabling leadership of headteachers and by national warrants for exercising leadership. The study demonstrates the value of the analytical framework and indicates that a cadre of teacher system leaders can be developed by attending to the interplay of professional development leadership and a wider system-orientated professional identity and by specific support to develop adaptive leadership capacities and skills.